On Constraint Manifolds of Spatial Closed Chains in Lorentzian Space

The objective of this paper is to explore the unexplored algebraic representations of the constraint manifolds associated with [Formula: see text] and [Formula: see text] spatial closed chains in Lorentzian space. Initially, we establish the structure equations of spatial closed chains by utilizing the structure equations for spatial open chains in Lorentzian space. Subsequently, we employ these structure equations to derive the algebraic expressions defining the constraint manifolds of [Formula: see text] and [Formula: see text] spatial closed chains in Lorentzian space, considering factors such as the causal character of the first link and the rotational axis of the cylindrical joint. This study will finalize all algebraic representations of constraint manifolds for spacelike and timelike mechanisms in Lorentzian space.